John McCarthy wrote:
> One might extend that idea to call science without potential
> application to human material prosperity `recreational science'.
>> Both ideas are wrong. Pure mathematics and other pure science are
> motivated by the collective curiosity of the human race.
>
Quine was not playing down the value of pure mathematics, I suppose. The
question is: does mathematics describe a mind-independent domain of reality?
The idea is that sciences do describe a mind-independent reality (i.e. this
universe), and since mathematics is an integral part of sciences, successes
of mathematical applications in sciences provide reasons for believing that
mathematics does so as well. Moreover, this will justify the basic
mathematical (set theoretical) axioms strictly required for mathematical
applications in sciences, not a specific mathematical theory based on those
axioms, which may never be applied just by chance. If an axiom is never
relevant to scientific applications, it is 'recreational' in the sense that
it is mathematicians' free creation and may not correspond to a mind
independent domain of reality (or at least we haven't got any reasons for
believing so).
In the current context of discussing Harvey Friedman's researches, the
question is: do they help to justify the mind-independent existence of large
cardinals? (Certainly, I am not saying that is this is Harvey's goal.) We
can put the question this way: Is Goedel's faith in Platonism supported by
the fact that some advanced axioms are required for scientific applications,
for deriving mundane truths, or is it solely based on mathematicians'
*intuitions*, without any support from scientific practices, or without any
relevance to any mundane matters?
For this matter, a Pi _{1}^{0} sentence A, independent of ZFC but derivable
from some T=ZFC+(a large cardinal axiom), may not be sufficient. Because, we
know that if T is an extension of ZFC (or just PRA) and A is Pi _{1}^{0},
then the following is provable finisitically (in PRA):
If T is consistent and T proves A, then A.
Therefore, if we know that T proves A, for us to accept A, we do not need to
accept the large cardinal axiom in T as it is understood Platonistically. We
only need to accept the consistency of T, which allows a mundane
interpretation and justification. For someone who is averse to Platonism,
our belief in the consistency of T is actually reached by inductions. That
is, after playing with those concepts and proofs in T (or ZFC) in our heads
for a long time, we come to believe that the proof patterns involved in
those axioms will not produce contradictions. This is a belief based on our
experiences in doing proofs and in reflecting upon our own such experiences.
We do not have to assume any mysterious mental faculty to perceive
mind-independent large cardinals, nor do we have to accept the literal
existence of large cardinals in order to accept A.